On Lelong Numbers of Positive Closed Currents on $\mathbb{P}^n$

Abstract

Let $T$ be a positive closed current of bidimension $(p,p)$ with unit mass on the complex projective space $\mathbb P^n$. For certain values of $\alpha$ and $\beta = \beta(p, \alpha)$ we show that if $T$ has enough points where the Lelong number is at least $\alpha$, then the upper level set $E_{\beta}^+ (T)$ of points where $T$ has Lelong number strictly larger than $\beta$ has certain geometric properties.